Groups, Geometry, and Dynamics

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Volume 11, Issue 2, 2017, pp. 613–647
DOI: 10.4171/GGD/410

Published online: 2017-06-26

Embedding mapping class groups into a finite product of trees

David Hume[1]

(1) UC Louvain, Louvain-La-Neuve, Belgium

We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are quasi-isometric to tree-graded spaces. Using this we prove that mapping class groups quasi-isometrically embed into a finite product of simplicial trees. In particular, these groups have finite Assouad–Nagata dimension, direct embeddings exhibiting $\ell^p$ compression exponent $1$ for all $p\geq 1$ and they quasi-isometrically embed into $\ell^1(\N)$. We deduce similar consequences for relatively hyperbolic groups whose parabolic subgroups satisfy such conditions.

In obtaining these results we also demonstrate that curve complexes of compact surfaces and coned-off graphs of relatively hyperbolic groups admit quasi-isometric embeddings into finite products of trees.

Keywords: Tree-graded space, quasi-tree, embeddings, mapping class group, curve complex

Hume David: Embedding mapping class groups into a finite product of trees. Groups Geom. Dyn. 11 (2017), 613-647. doi: 10.4171/GGD/410